I've never done it for birds, but I have for reptiles. I do try to get off as much of the bark as I can just because it's easiest to clean off any funguses or lichen that might be growing in the bark or underneath (you could probably leave a few scraps on the branch for your bird to peel off, but I wouldn't leave the entire branch that way). Then I soak it for a full 24 hours in a mixture of Dawn Soap and water in my bath tub, just to really disinfect it and wash away any nasty chemicals or tree sap that might still be around. After that, it gets rinsed off with fresh water until you think you got all the soap off. Then I put it in the oven at 250° for two hours (checking every ten minutes to make sure it's not burning or catching on fire) just to make sure it dries a bit faster. This is just optional, but I like to sand paper it down a bit at the end, makes any rough edges smoother.
Word of caution: if you think it was used around pesticides, I'd be extremely cautious. Usually I just don't use any wood if I don't know if pesticides were used around them, but I don't know who would spray down a willow tree so you're probably fine. Just a word of caution in the future if you find the perfect branch in someone's yard like I have.